Non-Local Equivariant Star Product on the Minimal Nilpotent Orbit

We construct a unique G-equivariant graded star product on the algebra $S(g)/I$ of polynomial functions on the minimal nilpotent coadjoint orbit \Omin of G where G is a complex simple Lie group and g\neq\sl_2(C). This strengthens the result of Arnal, Benamor, and Cahen. Our main result is to compute, for G classical, the star product of a momentum function $\mu_x$ with any function f. We find \mu_x\star f=\mu_xf+\half\{\mu_x,f\}t+\Lambda^x(f)t^2. For \g different from sp_n(\C), $\Lambda^x$ is not a differential operator. Instead \Lamda^x is the left quotient of an explicit order 4 algebraic differential operator $D^x$ by an order 2 invertible diagonalizable operator. Precisely, $\Lambda^x=-{1/4}\frac{1}{E'(E'+1)}D^x$ where $E'$ is a positive shift of the Euler vector field. Thus $\mu_x\star f$ is not local in f. Using $\star$ we construct a positive definite hermitian inner product on $Sg/I$. The Hilbert space completion of $Sg/I$ is then a unitary representation of $G$. This quantizes \Omin in the sense of geometric quantization and the orbit method.Comment: latex file, 13 pages. In this new version we use the star product to construct a unitary representation attached to the orbi

Projective modules over non-commutative tori: classification of modules with constant curvature connection

We study finitely generated projective modules over noncommutative tori. We prove that for every module $E$ with constant curvature connection the corresponding element $[E]$ of the K-group is a generalized quadratic exponent and, conversely, for every positive generalized quadratic exponent $\mu$ in the K-group one can find such a module $E$ with constant curvature connection that $[E] = \mu$. In physical words we give necessary and sufficient conditions for existence of 1/2 BPS states in terms of topological numbers.Comment: Latex. Misprints correcte

Fedosov's quantization of semisimple coadjoint orbits

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995.Includes bibliographical references (leaves 49-50).by Alexander Astashkevich.Ph.D

De Rham Cohomology of the Supermanifolds and Superstring BRST Cohomology

We show that the BRST operator of Neveu-Schwarz-Ramond superstring is closely related to de Rham differential on the moduli space of decorated super-Riemann surfaces P. We develop formalism where superstring amplitudes are computed via integration of some differential forms over a section of P over the super moduli space M. We show that the result of integration does not depend on the choice of section when all the states are BRST physical. Our approach is based on the geometrical theory of integration on supermanifolds of which we give a short review.Comment: 6 page

Quantum cohomology of partial flag manifolds

We compute the quantum cohomology rings of the partial flag manifolds F_{n_1\cdots n_k}=U(n)/(U(n_1)\times \cdots \times U(n_k)). The inductive computation uses the idea of Givental and Kim. Also we define a notion of the vertical quantum cohomology ring of the algebraic bundle. For the flag bundle F_{n_1\cdots n_k}(E) associated with the vector bundle E this ring is found.Comment: 33 page

Differential topology, infinite-dimensional Lie algebras, and applications: D. B. Fuchs' 60th anniversary collection

This volume presents contributions by leading experts in the field. The articles are dedicated to D. B. Fuchs on the occasion of his 60th birthday. Contributors to the book were directly influenced by Professor Fuchs and include his students, friends, and professional colleagues. In addition to their research, they offer personal reminicences about Professor Fuchs, giving insight into the history of Russian mathematics. The main topics addressed in this unique work are infinite-dimensional Lie algebras with applications (vertex operator algebras, conformal field theory, quantum integrable sys

Projective modules over non-commutative tori: classification of modules with constant curvature connection

We study finitely generated projective modules over noncommutative tori. We prove that for every module $E$ with constant curvature connection the corresponding element $[E]$ of the K-group is a generalized quadratic exponent and, conversely, for every positive generalized quadratic exponent $\mu$ in the K-group one can find such a module $E$ with constant curvature connection that $[E] = \mu$. In physical words we give necessary and sufficient conditions for existence of 1/2 BPS states in terms of topological numbers

Renaissance

Abstract. We obtain sharp bounds on mixing time of random walks on nilpotent groups, with Hall bases as generating sets